A Poisson likelihood approach to fake lepton estimation with the matrix method

Many high-energy physics analyses require the presence of leptons from $W$, $Z$, or $H$ boson decay. For these analyses, signatures that mimic such leptons present a `fake lepton' background that must be estimated. Since the magnitude of this background depends strongly upon details of the detector response, it can be difficult to estimate with simulation. One data-driven approach is the `matrix method', in which two categories of leptons are defined (`loose' and `tight'), with the tight category being a subset of the loose category. Using the populations of leptons in each category in the analysis sample, and the efficiencies for both real and fake leptons in the loose category to satisfy the criteria for the tight category, the fake background yield can be estimated. This paper describes a Poisson likelihood implementation of the matrix method, which provides a more precise, reliable, and robust estimate of the fake background yield compared to an analytic solution. This implementation also provides a reliable estimate of the background for cases in which the analysis selection permits more loose leptons than tight leptons, potentially allowing for greater selection efficiency.